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Inertial-thermal governed vapor bubble growth in highly superheated liquid

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Abstract

A generalized inertial thermal model for the vapor bubble growth in an infinite volume of uniformly superheated liquid is developed. An approximate analytical solution to the problem is obtained. The solution provides exact asymptotic limiting processes over all defining problem parameters. The physically correct description of experimental data related to vapor bubble growth in the near-spinodal region, which were previously considered as paradoxical, is presented for the first time.

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Fig. 1a–d
Fig. 2
Fig. 3
Fig. 4
Fig. 5

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Notes

  1. In the original papers [14–18], instead of the Plesset–Zwick equation, the universal Prosperitti–Plesset relationship [3] is applied, which takes into account the surface tension and radius of the critical vapor nucleus and uses Eq. 10 as the asymptotic form of the thermal mechanism. In the conditions under consideration, these equations practically coincide to one another.

  2. It is worth noting that all experimental and numerical studies of the vapor bubble growth with which the solution of Mikic et al. [9] was compared had been performed at relatively low superheating of liquid (N<<1).

  3. It is interesting to note that the experimental points of Frost and Sturtevant [17] were obtained for the region of rather long values of the dimensionless time and, therefore, correspond to the ultimate case that is described by asymptotic relationship (22) (thermal energy mechanism).

Abbreviations

a :

thermal diffusivity of liquid

C p :

isobaric specific heat of liquid

Ja:

Jakob number (=ρLcpΔT/ρVL)

l :

length scale (=a/UJa2)

L :

latent heat of phase transition

m :

growth constant \(\left( { = {R/ {\sqrt {at} }}}\right)\)

N :

parameter of superheating of liquid (=cpΔT/L)

n :

modified growth constant \(\left( { = \varepsilon m/\sqrt {12} } \right)\)

p :

pressure

p 0 :

saturation pressure corresponding to the temperature of liquid at infinity

p ∞ :

static pressure in a system

Δp:

pressure difference between liquid and the bubble surface (=p−p∞)

Δp0:

p0−p∞

\(\dot q\) :

heat flux density at the interface

R :

vapor bubble radius

\(\dot R\) :

vapor bubble growth rate (=dR/dt)

\(\ddot R\) :

d2R/dt2

R g :

individual gas constant

r :

dimensionless radius of a vapor bubble (=R/l)

T :

temperature

T ∞ :

temperature of liquid at infinity (far from a bubble)

T s :

saturation temperature at static pressure in a system

ΔT:

temperature difference between liquid at infinity and a vapor bubble (T∞-T)

ΔT0:

(T∞−Ts)

t :

time

t 0 :

time scale (=a/U2Ja2)

U :

velocity scale \(\left( { = \sqrt {{{2\Delta p_0 }/{3\rho _{\text{L}} }}} } \right)\)

U V :

velocity of vapor outflow from the interface (in the coordinate system, related to the interface surface)

U L :

velocity of liquid inflow to the interface (in the coordinate system, related to the interface surface)

U R :

velocity of motion of liquid at the interface (in the coordinate system, related to the center of a bubble)

u :

dimensionless bubble growth rate \(\left( { = {{\dot R}/ U}} \right)\)

β:

evaporation–condensation coefficient

η:

integration variable in Eq. 4

ɛ:

vapor/liquid density ratio (=ρV/ρL)

λ:

thermal conductivity of liquid

μ:

dynamic viscosity of liquid

ρV:

density of vapor

ρL:

density of liquid

τ:

dimensionless time (=t/t0)

ψ:

correction factor defined by Eq. 7 (interface permeability factor)

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Authors and Affiliations

  1. All-Russia Nuclear-Power Machine Building Research and Development Institute (VNIIAM), ul. Kosmonavta Volkova 6a, Moscow, 125171, Russia

    Alexandr A. Avdeev & Yuri B. Zudin

Authors
  1. Alexandr A. Avdeev
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  2. Yuri B. Zudin
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Correspondence to Alexandr A. Avdeev.

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Avdeev, A.A., Zudin, Y.B. Inertial-thermal governed vapor bubble growth in highly superheated liquid. Heat Mass Transfer 41, 855–863 (2005). https://doi.org/10.1007/s00231-004-0523-7

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